Modular Arithmetic Divisibility Criteria
Richard Singer · Mathematics Teacher Learning and Teaching PK-12 · 1970
There are well-known tests to determine when a positive integer k written in decimal notation is divisible by such numbers as 2, 3, 4, 5, 9, 10, or 11.1 These tests substitute for k some testing expression that can be checked more easily than k itself. For example, the testing expression for divisibility by 2 is the last digit of k, while the testing expression for 9 is the sum of the digits of k. In this paper we shall show how concise proofs of these tests can be obtained using the remainder function from the integers to the integers modulo n, obtain testing expressions for divisibility by other numbers such as 7 and 13, and examine how the integers modulo n can be used to determine divisiility in the absence of testing expressions.